Britain’s next prime minister will have a daunting in tray when he takes office this week.
Here’s a look at the challenges Jeremy Hunt or Boris Johnson will face when they step foot inside Number 10.
Finding a way to succeed where Theresa May failed by getting a Brexit deal through Parliament will be the most immediate political challenge.
Unless a snap general election is called to elect a new House of Commons, the incoming leader will face the same parliamentary difficulties that scuppered Mrs May’s attempts to build a coalition behind her proposals.
Alternatively, a new premier could pursue a no-deal policy and allow the UK to leave on October 31 without a formal agreement – although MPs may take steps to prevent that happening.
Either way, the new prime minister will have to find a way to reunite a Tory party which has splintered over the issue and counter the threat posed by Nigel Farage’s Brexit Party, which has sucked support away from the Conservatives.
Brexit has reignited the Scottish National Party’s push for independence.
Scotland voted to remain in the European Union and First Minister Nicola Sturgeon has said she wants another referendum on independence by 2021 if the country faces being taken out of the bloc.
In Northern Ireland, which also voted to remain in the EU in 2016, Sinn Fein has repeatedly called for a border poll to be conducted on whether there should be reunification with the Republic of Ireland.
Under the terms of the 1998 Good Friday peace agreement, the UK Government is obliged to call a vote on the constitutional issue if there is evidence of a change in public opinion in Northern Ireland in favour of Irish reunification.
A post-Brexit trade deal is one of the key prizes sought by the UK after leaving the EU, but negotiations are likely to run into difficulties over agricultural standards – with political rows over chlorine-washed chicken and hormone-treated beef – and Mr Trump’s “America first” approach to international affairs.
Managing the special relationship will be especially challenging for the next PM following a fraught few weeks of diplomatic drama which led to the resignation of the UK’s ambassador to the US, Sir Kim Darroch.
Unless Mrs May makes an appointment during her final days in office, Mr Hunt or Mr Johnson will soon have to make the key decision on who should be Britain’s representative in Washington.
The tensions caused by China’s rise as an economic and political powerhouse are felt across the West, with the row over whether to allow Huawei to contribute to the UK’s 5G network a symptom of wider unease.
Mr Trump’s US has adopted a tough public approach to China – banning Huawei and slapping tariffs on steel and other imports – while the UK has sought to build a “golden era” of relations with Beijing.
But pressure on the new PM from Washington, a final decision on Huawei and disputes over Beijing’s territorial claims in the South China Sea could lead to a rocky period for the UK-China relationship.
Fears of a slide into an all-out conflict between Iran and the West have heightened after a Royal Navy warship intervened to drive off Iranian patrol boats when they approached a British tanker as it sailed through the Strait of Hormuz.
It led to concerns the Iranians were trying to seize a UK ship in retaliation for the detention of the Iranian supertanker Grace 1, held in Gibraltar after being detained in an operation involving British Royal Marines.
The UK is at odds with its US allies over the Iran nuclear deal, but shares some of Washington’s concerns about Tehran’s wider activities in the Middle East.
With Mr Trump ramping up the US military presence in the area, the new prime minister could be forced to confront major decisions about war and peace early in their tenure.
However, Mr Johnson and Mr Hunt have both ruled out backing the US in any conflict against Iran.
The issue which, more than any other, derailed Mrs May’s 2017 general election campaign, her successor will have to come up with a system to cope with the rising costs of the UK’s ageing population.
A green paper setting out proposals on how to fund the system has been repeatedly delayed and the issue is politically toxic, with any suggestion of paying for care out of a person’s estate after they die liable to be condemned as a “death tax” by critics, while hiking income tax or national insurance could also be unpopular.
Successive governments have failed to get to grips with the nation’s housing shortage and the issue is likely to feature heavily in the Tory leadership contest.
The Government has a goal of building 300,000 new homes a year by the mid-2020s.